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Two things you need to know:
1) Most beaches are rocky, the pebbly ones are the best.
2) The best accommodation is rental apartments. You can do this direct from the owner, or through an agency or tourist office.

we were there in Sept 2008. Split is baeutiful, 2 nights...then we ferried to Hvar 2 nights, overnighted in Korcula then ferried to Dubrovnik for 3 nights. We didnt stay in any hotels , rented apts . It was fantastic! In Hvar you can rent smal motor boats to go to small outer islands with lovely beaches and restaurants...if you are interested I'll lookup the one we liked . Its a very interesting country.

Then I would only adjust my suggestion to say:
1) book places asap (you can find apartments online)
2) look at towns that are not so popular--skip Hvar town, but staying elsewhere on the island of Hvar may be just fine. (Hvar and its offshore islands have gorgeous beaches)
I liked the beaches best on Vis, but you would have to rent a car to get to them.
I suggest researching the stretch south of Split, as well as the Peljesac peninsula--possibly less crowds there.